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From the highly compressible Navier-Stokes equations to the porous medium equation - Rate of convergence

Haspot, B; Zatorska, E; (2016) From the highly compressible Navier-Stokes equations to the porous medium equation - Rate of convergence. Discrete and Continuous Dynamical Systems- Series A , 36 (6) pp. 3107-3123. 10.3934/dcds.2016.36.3107. Green open access

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Abstract

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to ε -1/2 for ε going to 0. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations -ρ ε converge to the unique solution to the porous medium equation [14, 13]. For viscosity coefficient μ(ρ ε ) = ρ ε α with α > 1, we obtain a rate of convergence of ρ ε in L ∞ (0,T;H -1 (ℝ)); for 1 < α ≤ 3/2 the solution ρ ε converges in L ∞ (0,T;L 2 (ℝ)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρ ε is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of ε.

Type: Article
Title: From the highly compressible Navier-Stokes equations to the porous medium equation - Rate of convergence
Open access status: An open access version is available from UCL Discovery
DOI: 10.3934/dcds.2016.36.3107
Publisher version: http://dx.doi.org/10.3934/dcds.2016.36.3107
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Compressible Navier-Stokes equations, porous medium equation, large Mach number
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10035985
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